integral of sqrt(x^2-2x^4)

Lösung

\int x^{2} - 2 x^{4} d x

- \frac{1}{6} (1 - 2 x^{2})^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{2} - 2 x^{4} d x

Vereinfache

x^{2} - 2 x^{4} : x^{2} 1 - 2 x^{2}

= \int x^{2} 1 - 2 x^{2} d x

x^{2} = (x),

angenommen

x \geq 0

= \int x 1 - 2 x^{2} d x

Wende U-Substitution an

= \int - \frac{u}{4} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{4} \cdot \int u d u

Wende die Potenzregel an

= - \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}}

Setze in

u = 1 - 2 x^{2}

ein

= - \frac{1}{4} \cdot \frac{2}{3} (1 - 2 x^{2})^{\frac{3}{2}}

Vereinfache

- \frac{1}{4} \cdot \frac{2}{3} (1 - 2 x^{2})^{\frac{3}{2}} : - \frac{1}{6} (1 - 2 x^{2})^{\frac{3}{2}}

= - \frac{1}{6} (1 - 2 x^{2})^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{6} (1 - 2 x^{2})^{\frac{3}{2}} + C

n = 0 \sum \infty \frac{2}{e ^{n}}

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